Am I allowed to wear an automatic watch on shabbos that works with the movement of my arm?
If the watch is still working on its own, it is permitted. The reason is since you are not moving your arm in order to wind the watch, therefore the action is considered davar sheino miskaven, and even if it will automatically wind the watch, since the watch is made that way. Nevertheless, the winding of the watch is not needed or wanted on shabbos, because the watch will anyways work until after shabbos, from the movements that were done on Friday. Therefore it is a “psik reisha d’lo ichpas lei”. This is coupled with other permitted factors, one is that there are opinions that winding a watch is permitted, and although we are stringent in this regard, even so it would only be an issur d’rabonon, plus the fact that the movement of the persons hand would only be a grama, together with the fact that it is considered a backhanded way to wind the watch. Therefore we have two or more d’rabonons by a psik reisha d’lo neicha lei, and it is permitted.
Chelkas Yackov 1-75, Shevet Halevi 3 -97, Tzitz Eliezer 9-20, Mishneh Halachos 8-59, Shmiras Shabbos Khilchoso 28 ftnt. 57 in the name of R’ S. Z Auerbach zt”l.